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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Green's matrix ： ウィキペディア英語版 Green's matrix In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green. For instance, consider $x\text{'}=A(t)x+g(t)\backslash ,$ where $x\backslash ,$ is a vector and $A(t)\backslash ,$ is an $n\backslash times\; n\backslash ,$ matrix function of $t\backslash ,$, which is continuous for $t\backslash isin\; I,\; a\backslash le\; t\backslash le\; b\backslash ,$, where $I\backslash ,$ is some interval. Now let $x^1(t),\backslash ldots,x^n(t)\backslash ,$ be $n\backslash ,$ linearly independent solutions to the homogeneous equation $x\text{'}=A(t)x\backslash ,$ and arrange them in columns to form a fundamental matrix: :$X(t)\; =\; \backslash left(x^1(t),\backslash ldots,x^n(t)\; \backslash right).\backslash ,$ Now $X(t)\backslash ,$ is an $n\backslash times\; n\backslash ,$ matrix solution of $X\text{'}=AX\backslash ,$. This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation. Let $x\; =\; Xy\backslash ,$ be the general solution. Now, :$$ \begin x' & =X'y+Xy' \\ & = AXy+Xy' \\ & = Ax + Xy'. \end This implies $Xy\text{'}=g\backslash ,$ or $y\; =\; c+\backslash int\_a^t\; X^(s)g(s)\backslash ,ds\backslash ,$ where $c\backslash ,$ is an arbitrary constant vector. Now the general solution is $x=X(t)c+X(t)\backslash int\_a^t\; X^(s)g(s)\backslash ,ds.\backslash ,$ The first term is the homogeneous solution and the second term is the particular solution. Now define the Green's matrix The particular solution can now be written $x\_p(t)\; =\; \backslash int\_a^b\; G\_0(t,s)g(s)\backslash ,ds.\backslash ,$ ==External links== *(An example ) of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Green's matrix」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.